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and its apparent frequency decreases such that $$\frac{\nu'}{\nu}=\sqrt{1-\frac{v^{2}}{c^{2}}}$$

so $$\frac{\nu'}{\nu}=\frac{m}{m'}$$

but equating the energies $$ h\nu= mc^{2}$$

gives the inverse relationship $$\frac{\nu'}{\nu}=\frac{m'}{m}$$

Has this question already been addressed? Otherwise, thank you in advance if someone can point out my mistake.